Isomorphism between groups...

Hello!

I am trying to show that is an ismorphism,

defined by where

where U(n) is the multiplicative group of units (refresher: numbers relativly prime to n)

and is the external direct product.

so clearly is well defined

I have shown it is 1-1

So my problem is...

I am having trouble showing is it onto (to show to that it is a bijection)

so let . So b is an arbitrary element of .

so what we need to do is find some such that

(we must find a x that maps to b... defn of onto, right?)

Keep in mind that we know that since

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So I thought maybe ?

then by the definition of alpha,

Now we just need to get rid of the t in the first component and s in the second component to map to that arbitrary element

Yikes sorry for all the description. All help would be much appriciated, this actually stumped our class for a few minutes at the end of class today!

Thanks! in advance!